Mathematically for (a), you need to prove that each side of the quadrilateral is equal in magnitude and that opposing sides of the quadrilateral are parallel and perpendicular to the other sides.

Length is easy

length = √[(x2-x1)² + (y2-y1)²]

So the length from (-3,1) to (-1,4) is √[(-1--3)² + (4-1)²] = √[4 + 9] = √13

And the length from (-3,1) to (0,-1) is √[(0--3)² + (-1-1)²] = √[9 + 4] = √13

To prove they are perpendicular to each other just find the gradients of each side and two of the lines will have gradient m whilst the other two sides will have gradient -1/m

The gradient can be found using (y2 - y1)/(x2 - x1)

So the gradient from (-3,1) to (-1,4) is => (4-1)/(-1-1) = -3/2

And the gradient from (-3,1) to (0,-1) is => (-1-1)/(0--3) = 2/3

So just do this for the other two sides, proving their lengths are root 13, and their gradients are m, -1/m, m and -1/m.

(b) is similar. Work out the equations of the diagonals using y = mx + c, filling in the values of y and x for each pair of diagnol coordinates. Then show that the gradient, m, for one diagonal is perpendicular to the other diagnol with gradient -1/m.

(c) is just working out the length of the diagonals using the formulae i gave you in part (a). Simple enough. The values of these lines should come out as √26 since this is what pythagorus theorem suggests.

Using pythagorus theorem => a² = b² + c²

a² = (√13)² + (√13)²

a² = 13 + 13 = 26

a = √26

Length = √[(2 - 1)² + (2 - -3)²] = √[1 + 25] = √26